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Theorem noinfbnd1lem4 27771
Description: Lemma for noinfbnd1 27774. If 𝑈 is a prolongment of 𝑇 and in 𝐵, then (𝑈‘dom 𝑇) is not undefined. (Contributed by Scott Fenton, 9-Aug-2024.)
Hypothesis
Ref Expression
noinfbnd1.1 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
noinfbnd1lem4 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ ∅)
Distinct variable groups:   𝐵,𝑔,𝑢,𝑣,𝑥,𝑦   𝑣,𝑈   𝑔,𝑉   𝑥,𝑈,𝑦
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑈(𝑢,𝑔)   𝑉(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem noinfbnd1lem4
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 simpl1 1192 . . . . . . . 8 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → ¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2 simpl2 1193 . . . . . . . 8 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝐵 No 𝐵𝑉))
3 simprl 771 . . . . . . . 8 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑤𝐵)
4 simpl3 1194 . . . . . . . . 9 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇))
5 simp2l 1200 . . . . . . . . . . . . 13 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝐵 No )
65sselda 3983 . . . . . . . . . . . 12 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤𝐵) → 𝑤 No )
7 simp3l 1202 . . . . . . . . . . . . . 14 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝑈𝐵)
85, 7sseldd 3984 . . . . . . . . . . . . 13 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝑈 No )
98adantr 480 . . . . . . . . . . . 12 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤𝐵) → 𝑈 No )
10 sltso 27721 . . . . . . . . . . . . 13 <s Or No
11 soasym 5625 . . . . . . . . . . . . 13 (( <s Or No ∧ (𝑤 No 𝑈 No )) → (𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤))
1210, 11mpan 690 . . . . . . . . . . . 12 ((𝑤 No 𝑈 No ) → (𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤))
136, 9, 12syl2anc 584 . . . . . . . . . . 11 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤𝐵) → (𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤))
1413impr 454 . . . . . . . . . 10 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → ¬ 𝑈 <s 𝑤)
153, 14jca 511 . . . . . . . . 9 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤𝐵 ∧ ¬ 𝑈 <s 𝑤))
16 noinfbnd1.1 . . . . . . . . . 10 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
1716noinfbnd1lem2 27769 . . . . . . . . 9 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑤𝐵 ∧ ¬ 𝑈 <s 𝑤))) → (𝑤 ↾ dom 𝑇) = 𝑇)
181, 2, 4, 15, 17syl112anc 1376 . . . . . . . 8 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤 ↾ dom 𝑇) = 𝑇)
1916noinfbnd1lem3 27770 . . . . . . . 8 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑤𝐵 ∧ (𝑤 ↾ dom 𝑇) = 𝑇)) → (𝑤‘dom 𝑇) ≠ 1o)
201, 2, 3, 18, 19syl112anc 1376 . . . . . . 7 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤‘dom 𝑇) ≠ 1o)
2120neneqd 2945 . . . . . 6 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → ¬ (𝑤‘dom 𝑇) = 1o)
2221expr 456 . . . . 5 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤𝐵) → (𝑤 <s 𝑈 → ¬ (𝑤‘dom 𝑇) = 1o))
23 imnan 399 . . . . 5 ((𝑤 <s 𝑈 → ¬ (𝑤‘dom 𝑇) = 1o) ↔ ¬ (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1o))
2422, 23sylib 218 . . . 4 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤𝐵) → ¬ (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1o))
2524nrexdv 3149 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → ¬ ∃𝑤𝐵 (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1o))
26 breq2 5147 . . . . . . 7 (𝑥 = 𝑈 → (𝑦 <s 𝑥𝑦 <s 𝑈))
2726rexbidv 3179 . . . . . 6 (𝑥 = 𝑈 → (∃𝑦𝐵 𝑦 <s 𝑥 ↔ ∃𝑦𝐵 𝑦 <s 𝑈))
28 simpl1 1192 . . . . . . 7 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
29 dfral2 3099 . . . . . . . 8 (∀𝑥𝐵𝑦𝐵 𝑦 <s 𝑥 ↔ ¬ ∃𝑥𝐵 ¬ ∃𝑦𝐵 𝑦 <s 𝑥)
30 ralnex 3072 . . . . . . . . 9 (∀𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ ¬ ∃𝑦𝐵 𝑦 <s 𝑥)
3130rexbii 3094 . . . . . . . 8 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ ∃𝑥𝐵 ¬ ∃𝑦𝐵 𝑦 <s 𝑥)
3229, 31xchbinxr 335 . . . . . . 7 (∀𝑥𝐵𝑦𝐵 𝑦 <s 𝑥 ↔ ¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
3328, 32sylibr 234 . . . . . 6 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ∀𝑥𝐵𝑦𝐵 𝑦 <s 𝑥)
34 simpl3l 1229 . . . . . 6 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → 𝑈𝐵)
3527, 33, 34rspcdva 3623 . . . . 5 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ∃𝑦𝐵 𝑦 <s 𝑈)
36 breq1 5146 . . . . . 6 (𝑦 = 𝑤 → (𝑦 <s 𝑈𝑤 <s 𝑈))
3736cbvrexvw 3238 . . . . 5 (∃𝑦𝐵 𝑦 <s 𝑈 ↔ ∃𝑤𝐵 𝑤 <s 𝑈)
3835, 37sylib 218 . . . 4 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ∃𝑤𝐵 𝑤 <s 𝑈)
39 simpl2l 1227 . . . . . . . . . 10 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → 𝐵 No )
4039adantr 480 . . . . . . . . 9 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝐵 No )
41 simprl 771 . . . . . . . . 9 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑤𝐵)
4240, 41sseldd 3984 . . . . . . . 8 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑤 No )
4334adantr 480 . . . . . . . . 9 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑈𝐵)
4440, 43sseldd 3984 . . . . . . . 8 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑈 No )
45 simpl2 1193 . . . . . . . . . . 11 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → (𝐵 No 𝐵𝑉))
4616noinfno 27763 . . . . . . . . . . 11 ((𝐵 No 𝐵𝑉) → 𝑇 No )
4745, 46syl 17 . . . . . . . . . 10 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → 𝑇 No )
4847adantr 480 . . . . . . . . 9 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑇 No )
49 nodmon 27695 . . . . . . . . 9 (𝑇 No → dom 𝑇 ∈ On)
5048, 49syl 17 . . . . . . . 8 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → dom 𝑇 ∈ On)
51 simpll1 1213 . . . . . . . . . 10 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → ¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
52 simpll2 1214 . . . . . . . . . 10 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝐵 No 𝐵𝑉))
53 simpll3 1215 . . . . . . . . . 10 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇))
54 simprr 773 . . . . . . . . . . . 12 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑤 <s 𝑈)
5542, 44, 12syl2anc 584 . . . . . . . . . . . 12 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤))
5654, 55mpd 15 . . . . . . . . . . 11 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → ¬ 𝑈 <s 𝑤)
5741, 56jca 511 . . . . . . . . . 10 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤𝐵 ∧ ¬ 𝑈 <s 𝑤))
5851, 52, 53, 57, 17syl112anc 1376 . . . . . . . . 9 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤 ↾ dom 𝑇) = 𝑇)
59 simpl3r 1230 . . . . . . . . . 10 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → (𝑈 ↾ dom 𝑇) = 𝑇)
6059adantr 480 . . . . . . . . 9 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑈 ↾ dom 𝑇) = 𝑇)
6158, 60eqtr4d 2780 . . . . . . . 8 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤 ↾ dom 𝑇) = (𝑈 ↾ dom 𝑇))
62 simplr 769 . . . . . . . 8 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑈‘dom 𝑇) = ∅)
63 nogt01o 27741 . . . . . . . 8 (((𝑤 No 𝑈 No ∧ dom 𝑇 ∈ On) ∧ ((𝑤 ↾ dom 𝑇) = (𝑈 ↾ dom 𝑇) ∧ 𝑤 <s 𝑈) ∧ (𝑈‘dom 𝑇) = ∅) → (𝑤‘dom 𝑇) = 1o)
6442, 44, 50, 61, 54, 62, 63syl321anc 1394 . . . . . . 7 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤‘dom 𝑇) = 1o)
6564expr 456 . . . . . 6 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ 𝑤𝐵) → (𝑤 <s 𝑈 → (𝑤‘dom 𝑇) = 1o))
6665ancld 550 . . . . 5 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ 𝑤𝐵) → (𝑤 <s 𝑈 → (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1o)))
6766reximdva 3168 . . . 4 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → (∃𝑤𝐵 𝑤 <s 𝑈 → ∃𝑤𝐵 (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1o)))
6838, 67mpd 15 . . 3 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ∃𝑤𝐵 (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1o))
6925, 68mtand 816 . 2 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → ¬ (𝑈‘dom 𝑇) = ∅)
7069neqned 2947 1 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  cun 3949  wss 3951  c0 4333  ifcif 4525  {csn 4626  cop 4632   class class class wbr 5143  cmpt 5225   Or wor 5591  dom cdm 5685  cres 5687  Oncon0 6384  suc csuc 6386  cio 6512  cfv 6561  crio 7387  1oc1o 8499   No csur 27684   <s cslt 27685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-riota 7388  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689
This theorem is referenced by:  noinfbnd1lem5  27772  noinfbnd1lem6  27773
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